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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>
<a NAME="E1:30"/>
(  <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="rlvect_1.html#V6">right_complementable</a> &amp;  <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="clvect_1.html#V2">ComplexLinearSpace-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>



<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:30_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>





<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:30_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)</span>)</span> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)</span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T7" target="_self">Th7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T12" target="_self">Th12</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:30_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)ex <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

<br/>.= 
 <a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:30_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>





<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) ;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="cfuncdom.html#E23"><i><font color="Green">Lemma20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:30_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="complex1.html#K8">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>





<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:30_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="complex1.html#K9">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cfuncdom.html#T15" target="_self">Th15</a></i>;<br/>

<b>thus </b><a NAME="E8:30_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cfuncdom.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:30"/>
 <a href="clvect_1.html#G1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> </span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>
 ;<br/>


</div>
